Related papers: Higher Schwarzian, quasimodular forms and equivari…
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended…
In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential…
In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in…
We study various properties of quasimodular forms by using their connections with Jacobi-like forms and pseudodifferential operators. Such connections are made by identifying quasimodular forms for a discrete subgroup $\G$ of $SL(2, \bR)$…
We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group $\mathbb{H}_1$ and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood…
The role of Schwarzian derivative in the study of nonlinear ordinary differential equations is revisited. Solutions and invariances admitted by Painlev\'e XXV-Ermakov equation, Ermakov equation and third order linear equation in a normal…
Suppose that a transcendental meromorphic function in the plane has finitely many critical values, while its multiple points have bounded multiplicities, and its inverse function has finitely many transcendental singularities. Using the…
Lagrange introduced the notion of Schwarzian derivative and Thurston discovered its mysterious properties playing a role similar to that of curvature on Riemannian manifolds. Here we continue our studies on the development of the Schwarzian…
A decomposition of the Wiener measure based on its quasi-invariance under the group of diffeo- morphisms is proposed. As a result, functional integrals in the Schwarzian theory can be written as the Fourier transform of the integrals in a…
The Schr\" odinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring…
We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first…
This paper establishes a sharp Schwarz-Pick type inequality for real-valued invariant harmonic functions defined on the complex unit ball $\mathbb B^n$. The proof of this main result simultaneously provides a solution to a natural extension…
We start with introducing one of the most fundamental notions of differential geometry, Manifolds. We present some properties and constructions such as submanifolds, tangent spaces and the tangent map. Then we continue with introducing the…
There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its…
We study the Schwarzian derivative from a variational viewpoint. Firstly we show that the Schwarzian derivative defines a first integral of the Euler--Lagrange equation of a second order Lagrangian. Secondly, we show that the Schwarzian…
We derive the general rules of functional integration in the theories of Schwarzian type, thus completing the elaboration of Schwarzian functional integrals calculus initiated in \cite{(BShExact)}, \cite{(BShCorrel)}. Our approach is…
We define a higher-derivative generalization of Maxwell-Chern-Simons theory in $\mathcal{N}=1$ and $\mathcal{N}=2$ superspaces. In particular, the chosen higher-derivative operator is a polynomial function of the d'Alembertian of arbitrary…
We investigate the various types of weight raising and weight lowering operators on quasi-modular forms, or equivalently on Shimura's vector-valued modular forms involving symmetric power representations. We also present all the…
We study the meromorphic modular forms defined as sums of -k (k>1) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the…
Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss…